Conjugacy growth and hyperbolicity2. Conjugacy growth series in groups 3. Rivin’s conjecture for hyperbolic groups 4. Conjugacy representatives in acylindrically hyperbolic groups 2 / 48 Let G be a group with finite generating set X . I Denote by [g ] the conjugacy class of g ∈ G and by |g |c the conjugacy length of [g ], where |g |c is the length of the shortest h ∈ [g ], with respect to X . σG ,X (n) := ]{[g ] ∈ G | |g |c = n}. 3 / 48 Let G be a group with finite generating set X . I Denote by [g ] the conjugacy class of g ∈ G and by |g |c the conjugacy length of [g ], where |g |c is the length of the shortest h ∈ [g ], with respect to X . σG ,X (n) := ]{[g ] ∈ G | |g |c = n}. 3 / 48 Let G be a group with finite generating set X . I Denote by [g ] the conjugacy class of g ∈ G and by |g |c the conjugacy length of [g ], where |g |c is the length of the shortest h ∈ [g ], with respect to X . σG ,X (n) := ]{[g ] ∈ G | |g |c = n}. 3 / 48 I Guba-Sapir (2010): asymptotics of the conjugacy growth function for BS(1, n), the Heisenberg group on two generators, diagram groups, some HNN extensions. I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type ‘monsters’) groups of standard exponential growth should have exponential conjugacy growth. conjugacy growth for f.g. linear (non virt. nilpotent) groups. I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it is possible to construct groups with a prescribed conjugacy growth function. I Guba-Sapir (2010): asymptotics of the conjugacy growth function for BS(1, n), the Heisenberg group on two generators, diagram groups, some HNN extensions. I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type ‘monsters’) groups of standard exponential growth should have exponential conjugacy growth. conjugacy growth for f.g. linear (non virt. nilpotent) groups. I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it is possible to construct groups with a prescribed conjugacy growth function. I Guba-Sapir (2010): asymptotics of the conjugacy growth function for BS(1, n), the Heisenberg group on two generators, diagram groups, some HNN extensions. I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type ‘monsters’) groups of standard exponential growth should have exponential conjugacy growth. conjugacy growth for f.g. linear (non virt. nilpotent) groups. I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it is possible to construct groups with a prescribed conjugacy growth function. I Guba-Sapir (2010): asymptotics of the conjugacy growth function for BS(1, n), the Heisenberg group on two generators, diagram groups, some HNN extensions. I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type ‘monsters’) groups of standard exponential growth should have exponential conjugacy growth. conjugacy growth for f.g. linear (non virt. nilpotent) groups. I Hull-Osin (2013): conjugacy growth not quasi-isometry invariant. Also, it is possible to construct groups with a prescribed conjugacy growth function. Conjugacy growth in geometry A slight modification of the conjugacy growth function (including only the non-powers) appears in geometry: - counting the primitive closed geodesics of bounded length on a compact manifold M of negative curvature and exponential volume growth gives, via quasi-isometries, good (exponential) asymptotics for σ(n) for the fundamental group of M (Margulis, . . . ). Conjugacy growth in geometry A slight modification of the conjugacy growth function (including only the non-powers) appears in geometry: - counting the primitive closed geodesics of bounded length on a compact manifold M of negative curvature and exponential volume growth gives, via group of M (Margulis, . . . ). Conjugacy growth in geometry A slight modification of the conjugacy growth function (including only the non-powers) appears in geometry: - counting the primitive closed geodesics of bounded length on a compact manifold M of negative curvature and exponential volume growth gives, via quasi-isometries, good (exponential) asymptotics for σ(n) for the fundamental group of M (Margulis, . . . ). Let G be a group with finite generating set X . I The conjugacy growth series of G with respect to X records the number of conjugacy classes of every length. It is σ(G ,X )(z) := ∞∑ n=0 σ(G ,X )(n)zn, where σ(G ,X )(n) is the number of conjugacy classes of length n. 6 / 48 Conjecture (Rivin, 2000) If G hyperbolic, then the conjugacy growth series of G is rational if and only if G is virtually cyclic. Theorem (Antoln-C., 2015) If G is non-elementary hyperbolic, then the conjugacy growth series is transcendental. Theorem (C., Hermiller, Holt, Rees, 2014) Let G be a virtually cyclic group. Then the conjugacy growth series of G is rational. NB: Both results hold for all symmetric generating sets of G . 7 / 48 Conjecture (Rivin, 2000) If G hyperbolic, then the conjugacy growth series of G is rational if and only if G is virtually cyclic. Theorem (Antoln-C., 2015) If G is non-elementary hyperbolic, then the conjugacy growth series is transcendental. Theorem (C., Hermiller, Holt, Rees, 2014) Let G be a virtually cyclic group. Then the conjugacy growth series of G is rational. NB: Both results hold for all symmetric generating sets of G . 7 / 48 Conjecture (Rivin, 2000) If G hyperbolic, then the conjugacy growth series of G is rational if and only if G is virtually cyclic. Theorem (Antoln-C., 2015) If G is non-elementary hyperbolic, then the conjugacy growth series is transcendental. Theorem (C., Hermiller, Holt, Rees, 2014) Let G be a virtually cyclic group. Then the conjugacy growth series of G is rational. NB: Both results hold for all symmetric generating sets of G . 7 / 48 Conjugacy representatives In order to determine the conjugacy growth series, we need a set of minimal length conjugacy representatives, exactly one word w ∈ X ∗ such that 1. π(w) ∈ [g ], where π : X ∗ → G the natural projection, and 2. l(w) = |π(w)| = |π(w)|c is of minimal length in [g ], where I l(w) := word length of w ∈ X∗ I |g | = |g |X := the (group) length of g ∈ G with respect to X . 8 / 48 Conjugacy representatives In order to determine the conjugacy growth series, we need a set of minimal length conjugacy representatives, i.e. for each conjugacy class [g ] in G pick exactly one word w ∈ X ∗ such that 1. π(w) ∈ [g ], where π : X ∗ → G the natural projection, and 2. l(w) = |π(w)| = |π(w)|c is of minimal length in [g ], where I l(w) := word length of w ∈ X∗ I |g | = |g |X := the (group) length of g ∈ G with respect to X . 8 / 48 Conjugacy representatives In order to determine the conjugacy growth series, we need a set of minimal length conjugacy representatives, i.e. for each conjugacy class [g ] in G pick exactly one word w ∈ X ∗ such that 1. π(w) ∈ [g ], where π : X ∗ → G the natural projection, and 2. l(w) = |π(w)| = |π(w)|c is of minimal length in [g ], where I l(w) := word length of w ∈ X∗ I |g | = |g |X := the (group) length of g ∈ G with respect to X . 8 / 48 Conjugacy growth series in virt. cyclic groups: Z, Z2 ∗ Z2 In Z the conjugacy growth series is the same as the standard one: σ(Z,{1,−1})(z) = 1 + 2z + 2z2 + · · · = 1 + z 1− z . In Z2 ∗ Z2 a set of conjugacy representatives is 1, a, b, ab, abab, . . . , so σ(Z2∗Z2,{a,b})(z) = 1 + 2z + z2 + z4 + z6 · · · = 1 + 2z − 2z3 1− z2 . 9 / 48 Conjugacy growth series in virt. cyclic groups: Z, Z2 ∗ Z2 In Z the conjugacy growth series is the same as the standard one: σ(Z,{1,−1})(z) = 1 + 2z + 2z2 + · · · = 1 + z 1− z . In Z2 ∗ Z2 a set of conjugacy representatives is 1, a, b, ab, abab, . . . , so σ(Z2∗Z2,{a,b})(z) = 1 + 2z + z2 + z4 + z6 · · · = 1 + 2z − 2z3 1− z2 . 9 / 48 Conjugacy growth series in free groups: F2 = a, b Set a < b < a−1 < b−1 and choose as conjugacy representative the smallest shortlex rep. in each conjugacy class, so the language is {a±k , b±k , ab, ab−1, ba−1, a−1b−1, a2b,aba, · · · } 10 / 48 Asymptotics of conjugacy growth in the free group Idea: take all cyclically reduced words of length n, whose number is (2k − 1)n + 1 + (k − 1)[1 + (−1)n], and divide by n. Coornaert, 2005: For the free group Fk , the primitive (non-powers) conjugacy growth function is given by σp(n) ∼ (2k − 1)n+1 , h = log(2k − 1). In general, when powers are included, one cannot divide by n. 11 / 48 Asymptotics of conjugacy growth in the free group Idea: take all cyclically reduced words of length n, whose number is (2k − 1)n + 1 + (k − 1)[1 + (−1)n], and divide by n. Coornaert, 2005: For the free group Fk , the primitive (non-powers) conjugacy growth function is given by σp(n) ∼ (2k − 1)n+1 , h = log(2k − 1). In general, when powers are included, one cannot divide by n. 11 / 48 Asymptotics of conjugacy growth in the free group Idea: take all cyclically reduced words of length n, whose number is (2k − 1)n + 1 + (k − 1)[1 + (−1)n], and divide by n. Coornaert, 2005: For the free group Fk , the primitive (non-powers) conjugacy growth function is given by σp(n) ∼ (2k − 1)n+1 , h = log(2k − 1). In general, when powers are included, one cannot divide by n. 11 / 48 The conjugacy growth series in free groups • Rivin (2000, 2010): the conjugacy growth series of Fk is not rational: σ(z) = ∫ z 0 H(t) (1− x2)2 + ∞∑ d=1 Theorem (C. - Hermiller, 2012) For A, B finite groups with generating sets XA = A \ 1A, XB = B \ 1B , and A ∗ B with generating set X = XA ∪ XB . Then σ(A ∗ B,X ) is rational iff A = B = Z/2Z, i.e. A ∗ B = D∞. 13 / 48 Theorem (C. - Hermiller, 2012) For A, B finite groups with generating sets XA = A \ 1A, XB = B \ 1B , and A ∗ B with generating set X = XA ∪ XB . Then σ(A ∗ B,X ) is rational iff A = B = Z/2Z, i.e. A ∗ B = D∞. 13 / 48 A generating function f (z) is I rational if there exist polynomials P(z), Q(z) with integer coefficients such that f (z) = P(z) Q(z) ; I algebraic if there exists a polynomial P(x , y) with integer coefficients such that P(z , f (z)) = 0; I transcendental otherwise. Rivin’s conjecture ⇒ If G is non-elementary hyperbolic, then the conjugacy growth series σ is not rational. Proof. (Antoln-C., 2015) • Recall: σ(n) := ]{[g ] ∈ G | |g |c = n} is the strict conjugacy growth. • Let φ(n) := ]{[g ] ∈ G | |g |c ≤ n} be the cumulative conjugacy growth. 15 / 48 Rivin’s conjecture ⇒ If G is non-elementary hyperbolic, then the conjugacy growth series σ is not rational. Proof. (Antoln-C., 2015) • Recall: σ(n) := ]{[g ] ∈ G | |g |c = n} is the strict conjugacy growth. • Let φ(n) := ]{[g ] ∈ G | |g |c ≤ n} be the cumulative conjugacy growth. 15 / 48 Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper). Let G be a non-elementary word hyperbolic group. Then there are positive constants A,B and n0 such that A ehn ehn n for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|. MESSAGE:. The number of conjugacy classes in the ball of radius n is asymptotically the number of elements in the ball of radius n divided by n. 16 / 48 Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper). Let G be a non-elementary word hyperbolic group. Then there are positive constants A,B and n0 such that A ehn ehn n for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|. MESSAGE:. The number of conjugacy classes in the ball of radius n is asymptotically the number of elements in the ball of radius n divided by n. 16 / 48 Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper). Let G be a non-elementary word hyperbolic group. Then there are positive constants A,B and n0 such that A ehn ehn n for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|. Lemma (Flajolet: Trancendence of series based on bounds). Suppose there are positive constants A,B, h and an integer n0 ≥ 0 s.t. A ehn i=0 anz n is not algebraic. 17 / 48 Theorem [AC] (Conjugacy bounds based on Coornaert and Knieper). Let G be a non-elementary word hyperbolic group. Then there are positive constants A,B and n0 such that A ehn ehn n for all n ≥ n0, where h is the growth rate of G , i.e. ehn = |Ball(n)|. Lemma (Flajolet: Trancendence of series based on bounds). Suppose there are positive constants A,B, h and an integer n0 ≥ 0 s.t. A ehn i=0 anz n is not algebraic. 17 / 48 Bounds for the conjugacy growth Let φp(n) := ]{primitive [g ] ∈ G | |g |c ≤ n} be the primitive cumulative conjugacy growth. Theorem.(Coornaert and Knieper, GAFA 2002) Let G be a non-elementary word hyperbolic. Then there are positive constants A and n0 such that for all n ≥ n0 A ehn n ≤ φp(n). Theorem.(Coornaert and Knieper, IJAC 2004) Let G be a torsion-free non-elementary word hyperbolic group. Then there are positive constants B and n1 such that for all n ≥ n1 φp(n) ≤ B ehn Bounds for the conjugacy growth Let φp(n) := ]{primitive [g ] ∈ G | |g |c ≤ n} be the primitive cumulative conjugacy growth. Theorem.(Coornaert and Knieper, GAFA 2002) Let G be a non-elementary word hyperbolic. Then there are positive constants A and n0 such that for all n ≥ n0 A ehn n ≤ φp(n). Theorem.(Coornaert and Knieper, IJAC 2004) Let G be a torsion-free non-elementary word hyperbolic group. Then there are positive constants B and n1 such that for all n ≥ n1 φp(n) ≤ B ehn Bounds for the conjugacy growth Let φp(n) := ]{primitive [g ] ∈ G | |g |c ≤ n} be the primitive cumulative conjugacy growth. Theorem.(Coornaert and Knieper, GAFA 2002) Let G be a non-elementary word hyperbolic. Then there are positive constants A and n0 such that for all n ≥ n0 A ehn n ≤ φp(n). Theorem.(Coornaert and Knieper, IJAC 2004) Let G be a torsion-free non-elementary word hyperbolic group. Then there are positive constants B and n1 such that for all n ≥ n1 φp(n) ≤ B ehn Rivin’s conjecture ⇒ : Proof 1. Drop torsion requirement from upper bound of Coornaert and Knieper: (i) use the fact that there exists m <∞ such that all finite subgroups F ≤ G satisfy |F | ≤ m. ) cyclic permutations of a primitive conjugacy representative of length n correspond to different elements of length n in G . 2. Find conjugacy growth upper bound for all conjugacy classes, i.e. include the non-primitive classes in the count.
Rivin’s conjecture ⇒ : Proof 1. Drop torsion requirement from upper bound of Coornaert and Knieper: (i) use the fact that there exists m <∞ such that all finite subgroups F ≤ G satisfy |F | ≤ m. ) cyclic permutations of a primitive conjugacy representative of length n correspond to different elements of length n in G . 2. Find conjugacy growth upper bound for all conjugacy classes, i.e. include the non-primitive classes in the count.
1. Rivin’s conjecture for relatively hyperbolic groups? (a) we need sharp bounds for the standard growth function [Yang] √ (b) we need sharp bounds for the conjugacy growth function. 2. Rivin’s conjecture for acylindrically hyperbolic groups: Is the conjugacy growth series of a f.g. acylindrically hyperbolic group transcendental? 1. Rivin’s conjecture for relatively hyperbolic groups? (a) we need sharp bounds for the standard growth function [Yang] √ (b) we need sharp bounds for the conjugacy growth function. 2. Rivin’s conjecture for acylindrically hyperbolic groups: Is the conjugacy growth series of a f.g. acylindrically hyperbolic group transcendental? 1. Rivin’s conjecture for relatively hyperbolic groups? (a) we need sharp bounds for the standard growth function [Yang] √ (b) we need sharp bounds for the conjugacy growth function. 2. Rivin’s conjecture for acylindrically hyperbolic groups: Is the conjugacy growth series of a f.g. acylindrically hyperbolic group transcendental? 21 / 48 Formal languages and the Chomsky hierarchy Let X be a finite alphabet. A formal language over X is a set L ⊂ X∗ of words. regular unambiguous context-free context-free context-sensitive Formal languages and the Chomsky hierarchy Let X be a finite alphabet. A formal language over X is a set L ⊂ X∗ of words. regular unambiguous context-free context-free context-sensitive I The growth function fL : N→ N of L is: fL(n) = ]{w ∈ L | w of length n}. I The growth series of L is SL(z) = ∞∑ n=0 I Unambiguous context-free languages have ALGEBRAIC growth series. (Chomsky-Schutzenberger) I The growth function fL : N→ N of L is: fL(n) = ]{w ∈ L | w of length n}. I The growth series of L is SL(z) = ∞∑ n=0 I Unambiguous context-free languages have ALGEBRAIC growth series. (Chomsky-Schutzenberger) I The growth function fL : N→ N of L is: fL(n) = ]{w ∈ L | w of length n}. I The growth series of L is SL(z) = ∞∑ n=0 I Unambiguous context-free languages have ALGEBRAIC growth series. (Chomsky-Schutzenberger) Corollary. [AC] Let G be a non-elementary hyperbolic group, X a finite generating set and Lc any set of minimal length representatives of conjugacy classes. Then Lc is not regular. By Chomsky-Schuzenberger, Lc is not unambiguous context-free (UCF). 24 / 48 Corollary. [AC] Let G be a non-elementary hyperbolic group, X a finite generating set and Lc any set of minimal length representatives of conjugacy classes. Then Lc is not regular. By Chomsky-Schuzenberger, Lc is not unambiguous context-free (UCF). 24 / 48 Main Theorem [AC, 2015] Let G be an acylindrically hyperbolic group, X any finite generating set, and Lc be a set containing one minimal length representative of each conjugacy class. Then Lc is not unambiguous context-free, so not regular. 25 / 48 Main Theorem [AC, 2015] Let G be an acylindrically hyperbolic group, X any finite generating set, and Lc be a set containing one minimal length representative of each primitive conjugacy class/commensurating class. 26 / 48 (Dahmani, Guirardel, Osin, Hamenstadt, Bowditch, Fujiwara, Minasyan ...) I relatively…
